/usr/share/doc/octave/octave.html/Specialized-Solvers.html

<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Created by GNU Texinfo 6.5, http://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Specialized Solvers (GNU Octave)</title>

<meta name="description" content="Specialized Solvers (GNU Octave)">
<meta name="keywords" content="Specialized Solvers (GNU Octave)">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<link href="index.html#Top" rel="start" title="Top">
<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
<link href="Linear-Algebra.html#Linear-Algebra" rel="up" title="Linear Algebra">
<link href="Vectorization-and-Faster-Code-Execution.html#Vectorization-and-Faster-Code-Execution" rel="next" title="Vectorization and Faster Code Execution">
<link href="Functions-of-a-Matrix.html#Functions-of-a-Matrix" rel="prev" title="Functions of a Matrix">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.indentedblock {margin-right: 0em}
blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smalllisp {margin-left: 3.2em}
kbd {font-style: oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nolinebreak {white-space: nowrap}
span.roman {font-family: initial; font-weight: normal}
span.sansserif {font-family: sans-serif; font-weight: normal}
ul.no-bullet {list-style: none}
-->
</style>
<link rel="stylesheet" type="text/css" href="octave.css">


</head>

<body lang="en">
<a name="Specialized-Solvers"></a>
<div class="header">
<p>
Previous: <a href="Functions-of-a-Matrix.html#Functions-of-a-Matrix" accesskey="p" rel="prev">Functions of a Matrix</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Specialized-Solvers-1"></a>
<h3 class="section">18.5 Specialized Solvers</h3>
<a name="index-matrix_002c-specialized-solvers"></a>

<a name="XREFbicg"></a><dl>
<dt><a name="index-bicg"></a>: <em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>)</em></dt>
<dt><a name="index-bicg-1"></a>: <em><var>x</var> =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>P</var>)</em></dt>
<dt><a name="index-bicg-2"></a>: <em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>bicg</strong> <em>(<var>A</var>, <var>b</var>, &hellip;)</em></dt>
<dd><p>Solve <code>A x = b</code> using the Bi-conjugate gradient iterative method.
</p>
<ul class="no-bullet">
<li>- <var>rtol</var> is the relative tolerance, if not given or set to [] the
default value 1e-6 is used.

</li><li>- <var>maxit</var> the maximum number of outer iterations, if not given or
set to [] the default value <code>min (20, numel (b))</code> is used.

</li><li>- <var>x0</var> the initial guess, if not given or set to [] the default
value <code>zeros (size (b))</code> is used.
</li></ul>

<p><var>A</var> can be passed as a matrix or as a function handle or inline function
<code>f</code> such that <code>f(x, &quot;notransp&quot;) = A*x</code> and
<code>f(x, &quot;transp&quot;) = A'*x</code>.
</p>
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.  Both <var>M1</var>
and <var>M2</var> can be passed as a matrix or as a function handle or inline
function <code>g</code> such that <code>g(x, &quot;notransp&quot;) = M1 \ x</code> or
<code>g(x, &quot;notransp&quot;) = M2 \ x</code> and <code>g(x, &quot;transp&quot;) = M1' \ x</code> or
<code>g(x, &quot;transp&quot;) = M2' \ x</code>.
</p>
<p>If called with more than one output parameter
</p>
<ul class="no-bullet">
<li>- <var>flag</var> indicates the exit status:

<ul class="no-bullet">
<li>- 0: iteration converged to the within the chosen tolerance

</li><li>- 1: the maximum number of iterations was reached before convergence

</li><li>- 3: the algorithm reached stagnation
</li></ul>

<p>(the value 2 is unused but skipped for compatibility).
</p>
</li><li>- <var>relres</var> is the final value of the relative residual.

</li><li>- <var>iter</var> is the number of iterations performed.

</li><li>- <var>resvec</var> is a vector containing the relative residual at each
iteration.
</li></ul>


<p><strong>See also:</strong> <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFcgs">cgs</a>, <a href="#XREFgmres">gmres</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="#XREFqmr">qmr</a>.
</p>
</dd></dl>


<a name="XREFbicgstab"></a><dl>
<dt><a name="index-bicgstab"></a>: <em><var>x</var> =</em> <strong>bicgstab</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>)</em></dt>
<dt><a name="index-bicgstab-1"></a>: <em><var>x</var> =</em> <strong>bicgstab</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>P</var>)</em></dt>
<dt><a name="index-bicgstab-2"></a>: <em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>bicgstab</strong> <em>(<var>A</var>, <var>b</var>, &hellip;)</em></dt>
<dd><p>Solve <code>A x = b</code> using the stabilizied Bi-conjugate gradient iterative
method.
</p>
<ul class="no-bullet">
<li>- <var>rtol</var> is the relative tolerance, if not given or set to [] the
default value 1e-6 is used.

</li><li>- <var>maxit</var> the maximum number of outer iterations, if not given or
set to [] the default value <code>min (20, numel (b))</code> is used.

</li><li>- <var>x0</var> the initial guess, if not given or set to [] the default
value <code>zeros (size (b))</code> is used.
</li></ul>

<p><var>A</var> can be passed as a matrix or as a function handle or inline
function <code>f</code> such that <code>f(x) = A*x</code>.
</p>
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.  Both <var>M1</var>
and <var>M2</var> can be passed as a matrix or as a function handle or inline
function <code>g</code> such that <code>g(x) = M1 \ x</code> or <code>g(x) = M2 \ x</code>.
</p>
<p>If called with more than one output parameter
</p>
<ul class="no-bullet">
<li>- <var>flag</var> indicates the exit status:

<ul class="no-bullet">
<li>- 0: iteration converged to the within the chosen tolerance

</li><li>- 1: the maximum number of iterations was reached before convergence

</li><li>- 3: the algorithm reached stagnation
</li></ul>

<p>(the value 2 is unused but skipped for compatibility).
</p>
</li><li>- <var>relres</var> is the final value of the relative residual.

</li><li>- <var>iter</var> is the number of iterations performed.

</li><li>- <var>resvec</var> is a vector containing the relative residual at each
iteration.
</li></ul>


<p><strong>See also:</strong> <a href="#XREFbicg">bicg</a>, <a href="#XREFcgs">cgs</a>, <a href="#XREFgmres">gmres</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="#XREFqmr">qmr</a>.
</p>
</dd></dl>


<a name="XREFcgs"></a><dl>
<dt><a name="index-cgs"></a>: <em><var>x</var> =</em> <strong>cgs</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>)</em></dt>
<dt><a name="index-cgs-1"></a>: <em><var>x</var> =</em> <strong>cgs</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>P</var>)</em></dt>
<dt><a name="index-cgs-2"></a>: <em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>cgs</strong> <em>(<var>A</var>, <var>b</var>, &hellip;)</em></dt>
<dd><p>Solve <code>A x = b</code>, where <var>A</var> is a square matrix, using the
Conjugate Gradients Squared method.
</p>
<ul class="no-bullet">
<li>- <var>rtol</var> is the relative tolerance, if not given or set to [] the
default value 1e-6 is used.

</li><li>- <var>maxit</var> the maximum number of outer iterations, if not given or
set to [] the default value <code>min (20, numel (b))</code> is used.

</li><li>- <var>x0</var> the initial guess, if not given or set to [] the default
value <code>zeros (size (b))</code> is used.
</li></ul>

<p><var>A</var> can be passed as a matrix or as a function handle or inline
function <code>f</code> such that <code>f(x) = A*x</code>.
</p>
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.  Both <var>M1</var>
and <var>M2</var> can be passed as a matrix or as a function handle or inline
function <code>g</code> such that <code>g(x) = M1 \ x</code> or <code>g(x) = M2 \ x</code>.
</p>
<p>If called with more than one output parameter
</p>
<ul class="no-bullet">
<li>- <var>flag</var> indicates the exit status:

<ul class="no-bullet">
<li>- 0: iteration converged to the within the chosen tolerance

</li><li>- 1: the maximum number of iterations was reached before convergence

</li><li>- 3: the algorithm reached stagnation
</li></ul>

<p>(the value 2 is unused but skipped for compatibility).
</p>
</li><li>- <var>relres</var> is the final value of the relative residual.

</li><li>- <var>iter</var> is the number of iterations performed.

</li><li>- <var>resvec</var> is a vector containing the relative residual at
each iteration.
</li></ul>


<p><strong>See also:</strong> <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFbicg">bicg</a>, <a href="#XREFgmres">gmres</a>, <a href="#XREFqmr">qmr</a>.
</p></dd></dl>


<a name="XREFgmres"></a><dl>
<dt><a name="index-gmres"></a>: <em><var>x</var> =</em> <strong>gmres</strong> <em>(<var>A</var>, <var>b</var>, <var>m</var>, <var>rtol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>)</em></dt>
<dt><a name="index-gmres-1"></a>: <em><var>x</var> =</em> <strong>gmres</strong> <em>(<var>A</var>, <var>b</var>, <var>m</var>, <var>rtol</var>, <var>maxit</var>, <var>P</var>)</em></dt>
<dt><a name="index-gmres-2"></a>: <em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>gmres</strong> <em>(&hellip;)</em></dt>
<dd><p>Solve <code>A x = b</code> using the Preconditioned GMRES iterative method with
restart, a.k.a. PGMRES(m).
</p>
<ul class="no-bullet">
<li>- <var>rtol</var> is the relative tolerance,
if not given or set to [] the default value 1e-6 is used.

</li><li>- <var>maxit</var> is the maximum number of outer iterations, if not given or
set to [] the default value <code>min (10, numel (b) / restart)</code> is used.

</li><li>- <var>x0</var> is the initial guess,
if not given or set to [] the default value <code>zeros (size (b))</code> is used.

</li><li>- <var>m</var> is the restart parameter,
if not given or set to [] the default value <code>numel (b)</code> is used.
</li></ul>

<p>Argument <var>A</var> can be passed as a matrix, function handle, or inline
function <code>f</code> such that <code>f(x) = A*x</code>.
</p>
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.  Both <var>M1</var>
and <var>M2</var> can be passed as a matrix, function handle, or inline function
<code>g</code> such that <code>g(x) = M1\x</code> or <code>g(x) = M2\x</code>.
</p>
<p>Besides the vector <var>x</var>, additional outputs are:
</p>
<ul class="no-bullet">
<li>- <var>flag</var> indicates the exit status:

<dl compact="compact">
<dt>0 : iteration converged to within the specified tolerance</dt>
<dt>1 : maximum number of iterations exceeded</dt>
<dt>2 : unused, but skipped for compatibility</dt>
<dt>3 : algorithm reached stagnation (no change between iterations)</dt>
</dl>

</li><li>- <var>relres</var> is the final value of the relative residual.

</li><li>- <var>iter</var> is a vector containing the number of outer iterations and
total iterations performed.

</li><li>- <var>resvec</var> is a vector containing the relative residual at each
iteration.
</li></ul>


<p><strong>See also:</strong> <a href="#XREFbicg">bicg</a>, <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFcgs">cgs</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>, <a href="Iterative-Techniques.html#XREFpcr">pcr</a>, <a href="#XREFqmr">qmr</a>.
</p></dd></dl>


<a name="XREFqmr"></a><dl>
<dt><a name="index-qmr"></a>: <em><var>x</var> =</em> <strong>qmr</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>M1</var>, <var>M2</var>, <var>x0</var>)</em></dt>
<dt><a name="index-qmr-1"></a>: <em><var>x</var> =</em> <strong>qmr</strong> <em>(<var>A</var>, <var>b</var>, <var>rtol</var>, <var>maxit</var>, <var>P</var>)</em></dt>
<dt><a name="index-qmr-2"></a>: <em>[<var>x</var>, <var>flag</var>, <var>relres</var>, <var>iter</var>, <var>resvec</var>] =</em> <strong>qmr</strong> <em>(<var>A</var>, <var>b</var>, &hellip;)</em></dt>
<dd><p>Solve <code>A x = b</code> using the Quasi-Minimal Residual iterative method
(without look-ahead).
</p>
<ul class="no-bullet">
<li>- <var>rtol</var> is the relative tolerance, if not given or set to [] the
default value 1e-6 is used.

</li><li>- <var>maxit</var> the maximum number of outer iterations, if not given or
set to [] the default value <code>min (20, numel (b))</code> is used.

</li><li>- <var>x0</var> the initial guess, if not given or set to [] the default
value <code>zeros (size (b))</code> is used.
</li></ul>

<p><var>A</var> can be passed as a matrix or as a function handle or inline
function <code>f</code> such that <code>f(x, &quot;notransp&quot;) = A*x</code> and
<code>f(x, &quot;transp&quot;) = A'*x</code>.
</p>
<p>The preconditioner <var>P</var> is given as <code>P = M1 * M2</code>.  Both <var>M1</var>
and <var>M2</var> can be passed as a matrix or as a function handle or inline
function <code>g</code> such that <code>g(x, &quot;notransp&quot;) = M1 \ x</code> or
<code>g(x, &quot;notransp&quot;) = M2 \ x</code> and <code>g(x, &quot;transp&quot;) = M1' \ x</code> or
<code>g(x, &quot;transp&quot;) = M2' \ x</code>.
</p>
<p>If called with more than one output parameter
</p>
<ul class="no-bullet">
<li>- <var>flag</var> indicates the exit status:

<ul class="no-bullet">
<li>- 0: iteration converged to the within the chosen tolerance

</li><li>- 1: the maximum number of iterations was reached before convergence

</li><li>- 3: the algorithm reached stagnation
</li></ul>

<p>(the value 2 is unused but skipped for compatibility).
</p>
</li><li>- <var>relres</var> is the final value of the relative residual.

</li><li>- <var>iter</var> is the number of iterations performed.

</li><li>- <var>resvec</var> is a vector containing the residual norms at each
      iteration.
</li></ul>

<p>References:
</p>
<ol>
<li> R. Freund and N. Nachtigal, <cite>QMR: a quasi-minimal residual
method for non-Hermitian linear systems</cite>, Numerische Mathematik,
1991, 60, pp. 315-339.

</li><li> R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra,
V. Eijkhour, R. Pozo, C. Romine, and H. van der Vorst,
<cite>Templates for the solution of linear systems: Building blocks
for iterative methods</cite>, SIAM, 2nd ed., 1994.
</li></ol>


<p><strong>See also:</strong> <a href="#XREFbicg">bicg</a>, <a href="#XREFbicgstab">bicgstab</a>, <a href="#XREFcgs">cgs</a>, <a href="#XREFgmres">gmres</a>, <a href="Iterative-Techniques.html#XREFpcg">pcg</a>.
</p></dd></dl>



<hr>
<div class="header">
<p>
Previous: <a href="Functions-of-a-Matrix.html#Functions-of-a-Matrix" accesskey="p" rel="prev">Functions of a Matrix</a>, Up: <a href="Linear-Algebra.html#Linear-Algebra" accesskey="u" rel="up">Linear Algebra</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>



</body>
</html>
octave-doc 4.2.2-1ubuntu1 / usr / share / doc / octave / octave.html / Specialized-Solvers.html
